The Dirichlet norm and the norm of Szegő type

Saitoh, Saburou

doi:10.1090/S0002-9947-1979-0539923-5

The Dirichlet norm and the norm of Szegő type
HTML articles powered by AMS MathViewer

by Saburou Saitoh

Trans. Amer. Math. Soc. 254 (1979), 355-364

DOI: https://doi.org/10.1090/S0002-9947-1979-0539923-5

PDF | Request permission

Abstract:

Let S be a smoothly bounded region in the complex plane. Let $g(z,t)$ denote the Green’s function of S with pole at t. We show that \[ \iint _S {|f’(z){|^2} dx dy \leqslant \frac {1}{2}\int _{\partial S} {|f’(z){|^2}{{\left ( {\frac {{\partial g(z,t)}} {{\partial {n_z}}}} \right )}^{ - 1}}|dz|} }\] holds for any analytic function $f(z)$ on $S \cup \partial S$. This curious inequality is obtained as a special case of a much more general result.

References

N. Aronszajn, Theory of reproducing kernels, Trans. Amer. Math. Soc. 68 (1950), 337–404. MR 51437, DOI 10.1090/S0002-9947-1950-0051437-7
Saburou Saitoh, The kernel functions of Szegö type on Riemann surfaces, K\B{o}dai Math. Sem. Rep. 24 (1972), 410–421. MR 322161
Saburou Saitoh, The Rudin kernel and the extremal functions in Hardy classes, K\B{o}dai Math. Sem. Rep. 25 (1973), 37–47. MR 322162
Saburou Saitoh, On some completenesses of the Bergman kernel and the Rudin kernel, Pacific J. Math. 56 (1975), no. 2, 581–596. MR 422598, DOI 10.2140/pjm.1975.56.581
Saburou Saitoh, The exact Bergman kernel and the kernels of Szegö type, Pacific J. Math. 71 (1977), no. 2, 545–557. MR 454007, DOI 10.2140/pjm.1977.71.545
Saburou Saitoh, The Bergman norm and the Szegő norm, Trans. Amer. Math. Soc. 249 (1979), no. 2, 261–279. MR 525673, DOI 10.1090/S0002-9947-1979-0525673-8

Integral transform by the product of two Szegö kernel functions

Menahem Schiffer and Donald C. Spencer, Functionals of finite Riemann surfaces, Princeton University Press, Princeton, N. J., 1954. MR 0065652